Nonlinear Time Series: Nonparametric and Parametric Methods
Nonlinear Time Series: Nonparametric and Parametric Methods
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DOI：
10.1198/tech.2004.s746
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发表时间：
2004-02
期刊：
影响因子：
2.5
通讯作者：
Z.Q. John Lu
中科院分区：
文献类型：
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作者：
Z.Q. John Lu
增益函数（即，传递函数的绝对值）在这里被称为“调制传递函数”。第9章是统计理论的第一章，讨论了样本均值、加权平均值、样本方差和样本中位数在关于基本独立同分布（iid）数据（正态、二项式和与sinc函数平方成比例的特殊PDF）的各种分布假设下的统计特性。这些讨论总体上是好的（特别是对于平方sinc函数），尽管对于那些在数据分析方面没有什么经验的人来说，关于加权平均值的讨论可能过于抽象。第10章讨论的是概率定律的估计。忽略它令人困惑的介绍，本章以一个关于从iid数据估计PDF的正交函数方法的章节开始。后续部分讨论了涉及Karhunen-Loeve展开的变体，其中作者没有指出实际限制，即需要未知PDF的知识来构造正交函数。在这一节之后，本章开始了一个重大的出发点。数据不再被假设为RV的id实现，而是被认为是未知PDF的完全已知的函数（例如，我们的“数据”是未知PDF的均值和方差）。在这个框架下，作者讨论了一个类似贝叶斯的方法，称为“最大概率原理”，并将其与最大熵方法联系起来。从纯数学的角度来看，这本书很吸引人，所有最大熵原理的倡导者都应该阅读这本书。不幸的是，由于对数据构成的奇怪概念，这本书与PDF估计的统计问题只有很小的关系。第11章、第12章和第13章提供了关于显著性偏离预测艾德分布的卡方检验、均值的单样本t检验和方差相等的F检验的相当标准的讨论。第14章和第15章是对最小二乘理论和主成分分析的基本介绍，而第16章基本上是在评估硬币价格是否有偏差的背景下讨论贝叶斯统计理论。这本书的第17章，“估计方法介绍”，从最大似然估计，Cramer-Rao和Bhattacharyya下界以及贝叶斯估计理论的概述开始（尽管纯粹主义者会反对，因为缺乏关于各种结果所需条件的声明）。然而，就像在第10章中一样，这篇文章与大多数技术计量学读者所认为的统计估计理论有很大的不同。作者讨论了一种“基于费舍尔信息的方法”，该方法“旨在通过推导描述物理现象的波动方程来推导内斯该定律的真实概率定律”。本章的这一部分（有时带有哲学讨论的意味）显然是同一作者另一本书的梗概（Frieden，1998）。这本书缺少了许多在过去20年中在技术计量学中受到相当关注的主题（自举和非参数回归是两个突出的例子）。这种对旧技术的偏见可能是因为这本书是1983年出版的第三版。第1章至第16章的参考书目只包括了1990年以后出现的一些论文和书籍。（事实上，作者甚至没有费心更新几本书的参考资料，这些书本身现在已经在较新的版本中出版了。）第17章确实有许多参考资料，从过去的十年，似乎是主要的动机，这个新版本，但业主以前的版本可能会仔细看看序言，这个版本之前购买。我不愿意推荐概率，统计光学和数据测试作为初学者的教科书，因为它缺乏统计学中重要的新主题的覆盖面，并且因为太多古怪的非标准定义，这将使学生难以进入大量的统计和工程文献。然而，有一些真实的宝石包含在光学应用和作者提供的众多练习中。我会鼓励物理科学的教师和学生寻找这本书的一些具有挑战性的应用和统计问题。唉，为了与这本书的一般日期保持一致，作者指出，“选择问题的答案可以通过直接写信给作者来获得，并附上一个贴好邮票的回邮信封（约8 2 11英镑）“在电子邮件允许大量附件的时代，这是一种奇怪的方法。
gain function (i.e., the absolute value of the transfer function) is here termed the “modulation transfer function.” Chapter 9, the rst chapter on statistical theory, discusses the statistical properties of the sample mean, weighted averages, sample variance, and sample median under various distributional assumptions about the underlying independent and identically distributed (iid) data (normal, binomial, and a specialized PDF proportional to the square of a sinc function). The discussions are generally good (particularly for the squared sinc function), although the one on weighted averages might prove to be too abstract for those with little experience in data analysis. Chapter 10 is supposedly a discussion of the estimation of probability laws. Ignoring its confusing introduction, this chapter starts promisingly enough with a section on the orthogonal function approach to estimating PDFs from iid data. A follow-up section discusses a variation involving a Karhunen– Loeve expansion, in which the author fails to point out the practical limitation that knowledge of the unknown PDF is needed to construct the orthogonal functions. After this section, the chapter takes off on a major departure. The data are no longer assumed to be iid realizations of RVs, but rather are taken to be perfectly known functions of the unknown PDF (e.g., our “data” are the mean and variance of the unknown PDF). Under this framework, the author discusses a Bayesian-like approach called the “principle of maximum probability” and relates it to the maximum entropy approach. From a purely mathematical standpoint, this material is fascinating and should be required reading for all advocates of the maximum entropy principle; unfortunately, because of the bizarre notion of what constitutes data, it is only marginally relevant to the statistical problem of PDF estimation. Chapters 11, 12, and 13 provide fairly standard discussions on the chisquared test for signi cance departures from a prespeci ed distribution, the one-sample t test for a mean and the F test for equality of the variances. Chapters 14 and 15 are bare bones introductions to least squares theory and principal components analysis, whereas Chapter 16 is basically a discussion of Bayesian statistical theory in the context of assessing whether or not coin ips are biased. The book’s nal chapter, 17, “Introduction to Estimation Methods,” starts with a nice overview of maximum likelihood estimators, the Cramer–Rao and Bhattacharyya lower bounds, and Bayesian estimation theory (although purists will object because of the lack of statements about conditions needed for various results to hold). However, as in Chapter 10, the text then takes a major departure away from what most Technometrics readers would consider statistical estimation theory. The author discusses a “Fisher information-based approach” that “aims to nd the true probability law describing a physical phenomenon by deriving a wave equation that de nes the law.” This portion of the chapter (which at times has the avor of a philosophical discussion) is evidently a synopsis of another book by the same author (Frieden 1998). Missing from this book are many topics that have received considerable attention over the last 20 years in Technometrics (bootstrapping and nonparametric regression being two prominent examples). This bias toward older techniques is probably explained by the fact that this book is the third edition of a text that rst appeared in 1983. The reference lists for Chapters 1–16 include only a smattering of papers and books that have appeared after 1990. (Indeed, the author has not even bothered to update the references for several books that themselves are now out in newer editions.) Chapter 17 does have numerous references from the last decade and seems to be the main motivation for this new edition, but owners of previous editions might look carefully at the Preface to this edition before purchasing. I am reluctant to recommend Probability, Statistical Optics and Data Testing as a textbook for beginning students because of its lack of coverage of important new topics in statistics and because of too many quirky nonstandard de nitions that will make it dif cult for students to then jump into the bulk of the statistical and engineering literature. There are, however, some real gems contained in the optical applications and the numerous exercises that the author provides. I would encourage instructors and students of the physical sciences to seek out this book for some challenging applications and statistical problems. Alas, in keeping with the generally dated avor of the book, the author states that “answers to selected problems may be obtained by writing to the author directly, and enclosing a stamped, self-addressed envelope (about 8 2 £ 11 in) for return”—an odd approach in an age of e-mail permitting bulky attachments.